Optimizing Polynomial Approximations for Even Functions on Symmetric Intervals

[EvLer]

Homework Statement

find closest function a+bx³ to x² on the iterval [-1,1]
(we consider standard inner product (f,g) = integral(-1 to 1):fgdx

So, here is my attempt, but I got a suspicious result:

[(1,1) (1,x³)] [a]
[(1,x³) (x³,x³)]
=
[(1,x²)]
[x³, x²)]

then after I've done all the integration I came up with this:
[2 0] [a]
[0 2/7]
=
[1/3]
[0]
which means that x³ term disappears...

 

[Dick]

I can't help you fix it. Because you are completely correct. Why do you think that is suspicious? Find ways to check yourself.

 

[Dick]

You are approximating an even function over a symmetric interval. Can adding part of an odd function help you?